Bianchi IX geometry and the Einstein-Maxwell theory

A. M. Ghezelbash 1

Department of Physics and Engineering Physics,University of Saskatchewan,

Saskatoon, Saskatchewan S7N 5E2, Canada

Abstract

We construct numerical solutions to the higher-dimensional Einstein-Maxwell the-ory. The solutions are based on embedding the four dimensional Bianchi type IX spacein the theory. We find the solutions as superposition of two functions, which one ofthem can be found numerically. We show that the solutions in any dimensions, are al-most regular everywhere, except a singular point. We find that the solutions interpolatebetween the two exact analytical solutions to the higher dimensional Einstein-Maxwelltheory, which are based on Eguchi-Hanson type I and II geometries. Moreover, weconstruct the exact cosmological solutions to the theory, and study the properties ofthe solutions.

1E-Mail: masoud.ghezelbash@usask.ca

arX

iv:2

108.

0721

0v1

[gr

-qc]

16

Aug

202

1

1 Introduction

Exploring the different aspects of gravitational physics is possible through finding the newsolutions to gravity, especially coupled to the other fields, such as the electromagnetic field.In this regard, finding new solutions to higher dimensional gravitational theories revealsnew phenomena and possibilities, which may not exist in four dimensions. The discoveryof the black hole solutions in five dimensions with 3-sphere horizon [1], and the black ringswith S2 × S1 horizon [2], are just some of the rich variety of the black objects in five andhigher dimensions. Other relevant solutions in higher dimensional gravity coupled to thematter fields, are the dyonic and the solitonic solutions [3, 4], supergravity solutions [5]–[8],braneworld cosmologies [9], and string theory extended solutions [10]-[12].

To find new solutions to the higher-dimensional Einstein-Maxwell theory, we considerthe Bianchi type IX space, which is an exact solution to the four-dimensional Einstein’sequations. Different types of solutions to the Einstein-Maxwell theory were constructe, suchas solutions with the NUT charge [13], solitonic and dyonic solutions [3, 11], as well asbraneworld solutions [9]. Moreover, solutions to the extension of Einstein-Maxwell theorywith the axion field and Chern-Simons term, were constructed and studied extensively in[14].

In [15], the author found exact solutions to the eleven dimensional supergravity, wherethe membrane solutions have a transverse Bianchi type IX geometry. The solutions consistof analytically convoluted-like integrals of two functions, which depend on the transversedirections to the branes. Upon compactification over a compact dimension of transversegeometry, the solutions provide the fully localized intersecting type IIA branes. Due tothe self-duality of the transverse geometry in the constructed solutions, the compactifiedsolutions enjoy having eight supercharges.

In this article, inspired by the above-mentioned convoluted-like membrane solutions, wetry to find exact convoluted-like solutions to the Einstein-Maxwell theory. We note that oursolutions exist only in six or higher dimensions, as a result of the non-trivial convoluted-likesolutions.

We organize the article as follows: In section 2, we consider the physics of Bianchi typeIX spaces. In section 3, we present some numerical solutions to the Einstein-Maxwell theoryin six and higher dimensions, where the metric function can be written as the convoluted-likeintegral of two functions. In section 4, we present the second class of numerical solutionsto the Einstein-Maxwell theory in six and higher dimensions. The second class of solutionsare completely independent of the solutions in section 3. In section 5, we use the results ofsections 3 and 4, and explicitly construct some cosmological solutions to the Einstein-Maxwelltheory with positive cosmological constant, in six and higher dimensions. In section 6, weconsider the Bianchi type IX space with the special cases of the Bianchi parameter as k = 0and k = 1. We construct some exact solutions to the Einstein-Maxwell theory in six andhigher dimensions, where the radial function involves the Heun-C functions. We discuss thephysical properties of the solutions. We wrap up the article by the concluding remarks insection 7.

1

2 The Bianchi geometries

The classification of the homogeneous and isotropic spaces, is crucial to understand thecosmological models of the universe, as well as finding the theoretical models, which areconsistent with the experimental data. The first classification of the homogeneous spaces wasdone long time ago by Bianchi [16]. We know now that there are nine different homogeneousspaces [17], called Bianchi type I, · · · , IX. The Bianchi type IX has been used mainly incosmological models [18], supergravity theories [19, 20] and extensions of gravity [21].

The four-dimensional Bianchi Type IX metric, is given locally by the line element

ds2 = e2∑3i=1 Pi(η)dη2 +

3∑i=1

ePi(η)σ2i , (2.1)

where σi, i = 1, 2, 3 are three Maurer-Cartan forms, and Pi, i = 1, 2, 3 are three functions ofthe coordinate η. The Maurer-Cartan forms are given by

σ1 = dψ + cos θ dφ, (2.2)

σ2 = cosψ dθ + sinψ sin θ dφ, (2.3)

σ3 = − sinψ dθ + cosψ sin θ dφ, (2.4)

in terms of three coordinates θ, φ, and ψ of a unit S3. The line element (2.1) has an SU(2)isometry group. The metric satisfies exactly the vacuum Einstein’s equations, provided thefunctions Pi satisfy

2d2Pidη2

= e4Pi − (e2Pi+1 − e2Pi+2)2, (2.5)

as well as

4dP1

dη

dP2

dη+4

dP2

dη

dP3

dη+4

dP3

dη

dP1

dη= 2(e2P1+2P2+e2P2+2P3+e2P3+2P1)−(e4P1+e4P2+e4P3), (2.6)

where Pi+3 ≡ Pi in equation (2.5). Integrating the equations (2.5) and (2.6), we find

2dPidη

= −e2Pi + e2Pi+1 + e2Pi+2 − βiePi+1+Pi+2 , (2.7)

where βi, i = 1, 2, 3 are three integration constants, which satisfy βiβj = 2εijkβk. To find theBianchi type IX solutions, we consider βi = 0, i = 1, 2, 3, where the equations (2.7) can beintegrated to yield the well known solutions, in terms of the standard elliptic functions [19].After a change of coordinate from η to the spherical radial coordinate r, we find the metricfor the triaxial Bianchi type IX as

ds2BIX =dr2√F (r)

+r2√F (r)

4(

σ21

1− a41r4

+σ22

1− a42r4

+σ23

1− a43r4

), (2.8)

2

where ai, i = 1, 2, 3 are three integration constants, and the metric function F (r) is given by

F (r) =3∏i=1

(1− a4ir4

). (2.9)

We choose the three parameters as a1 = 0, a2 = 2kc and a3 = 2c, in term of 0 ≤ k ≤ 1and the constant c > 0. We note that a1 ≤ a2 ≤ a3. The metric is regular for all values ofthe radial coordinate r > 2c. The Ricci scalar for the Bianchi type IX space (2.8) is zeroand the Kretschmann invariant is given by

K =1649267441664c8

(2 ck − r)3 (8 c3k3 + 4 c2k2r + 2 ckr2 + r3)3 (4 c2 + r2)3 (4 c2 − r2)3 r12

×((

k8 − k4 + 1

16777216

)r24 − 3 c4k4 (k4 + 1) r20

1048576+

15 c8k8r16

65536− 5 c12k8 (k4 + 1) r12

2048

+3 c16k8 (k8 + 3 k4 + 1) r8

256− 3

16c20k12

(k4 + 1

)r4 + c24k16

). (2.10)

We notice the Kretschmann invariant (2.10) is regular everywhere, since r > 2c. Moreover,all components of the Ricci tensor are regular, too.

3 Embedding the Bianchi type IX space in D ≥ 6-

dimensional Einstein-Maxwell theory

We consider the D-dimensional Einstein-Maxwell theory

S =

∫dDx√−g(R− 1

4F 2), (3.1)

where Fµν = ∂νAµ − ∂µAν . We also consider the D-dimensional ansatz for the metric, as

ds2D = − dt2

H(r, x)2+H(r, x)

2D−3 (dx2 + x2dΩ2

D−6 + ds2BIX), (3.2)

where ds2BIX is given by (2.8) and dΩ2D−6 is the metric on a unit sphere SD−6, where D ≥ 6.

We also take the components of the Fµν

Ftr = − α

H(r, x)2∂H(r, x)

∂r, (3.3)

Ftx = − α

H(r, x)2∂H(r, x)

∂x, (3.4)

where α is a constant.

3

A very long calculation shows that all the Einstein’s and Maxwell’s field equations aresatisfied, if the metric function H(r, x) obeys the partial differential equation(

r9

256− 1

16c4(k4 + 1

)r5 + c8k4r

)(F (r)

∂2

∂r2H (r, x) +

√F (r)

∂2

∂x2H (r, x) + F ′ (r)

∂

∂rH (r, x)

)+ 7F (r)

(3 r8

1792− 5 c4 (k4 + 1) r4

112+ c8k4

)∂

∂rH (r, x) = 0. (3.5)

To solve the partial differential equation (3.5), we consider

H(r, x) = 1 + βR(r)X(x), (3.6)

where two functions R(r) and X(x) describe the separation of coordinates, and β is a con-stant. Plugging the equation (3.6) into equation (3.5), we find two ordinary differentialequations for the functions R(r) and X(x). The differential equation for the function X(x)is

xd2

dx2X(x) + (D − 6)

d

dxX(x)− g2xX(x) = 0, (3.7)

where g denote the separation constant. We find the solutions to (3.7) are given by

X(x) = x1IN(gx)

xN+ x2

KN(gx)

xN, (3.8)

where IN and KN are the modified Bessel functions of the first and second kind, respectively,and x1 and x2 are the integration constants and

N =D − 7

2. (3.9)

Moreover, we find the differential equation for the function R(r) as

(−256 c8k4 + 16 c4k4r4 + 16 c4r4 − r8

)r

d2

dr2R (r) +

(256 c8k4 + 16 c4k4r4 + 16 c4r4 − 3 r8

) d

drR (r)

− g2r5√

256 c8k4 − 16 c4k4r4 − 16 c4r4 + r8R (r) = 0. (3.10)

We note that the radial differential equation (3.10) is independent of the dimension D ofspace-time. Tough we can’t find any analytic solutions for the equation (3.10), however wetry to find the analytic solutions to the differential equation (3.10) in asymptotic regionr →∞. In the limit of r →∞, the equation (3.10) reduces to

rd2

dr2R(r) + 3

d

drR(r) + rg2R(r) = 0. (3.11)

The exact solutions to equation (3.11) are given by

R(r) = r1J1(gr)

r+ r2

Y1(gr)

r, (3.12)

4

for r →∞, where J1 and Y1 are the Bessel functions of the first and second kind, respectively.In figure 3.1, we plot the behaviour of the R(r) where r → ∞. We notice the asymptoticradial function monotonically and periodically approaches zero, as r →∞.

Figure 3.1: The radial function R(r) for large values of r, where we set r1 = 1, r2 = 0 (left),and r1 = 0, r2 = 1 (right) and g = 1.

Furnished by the asymptotic behaviour of the radial function, we solve numerically theradial differential equation (3.10) for 0 < k < 1. In figure 3.2, we plot the numerical solutionsfor the radial function R(r), where we set k = 1

2, c = 1 and g = 2. We notice from the figure

3.2 that the radial function becomes divergent as r → 2, and decays rapidly as r → ∞, inagreement with the asymptotic solutions (3.12) and figure 3.1.

Figure 3.2: The numerical solution for the radial function R(r), where we set k = 12, c = 1

and g = 2.

5

The general structure of the radial function is the same for other values of the Bianchiparameter c. The divergent behaviour of the radial function happens at r → 2c and theradial function decays rapidly for r →∞.

Moreover, in figure 3.3, we plot the numerical solutions for the radial function R(r),where we set k = 1

4and k = 3

4with c = 1 and g = 2.

Figure 3.3: The numerical solutions for the radial function R(r), where we set k = 14

(left)and k = 3

4(right) with c = 1, g = 2.

Tough the figures 3.2 and 3.3 are quite similar, however they have subtle dependenceon the Bianchi parameter k. In figure 3.4, we plot three radial functions, over a smallinterval of r, for k = 1

4, 1

2and 3

4. As we notice from figure 3.4, the radial function R(r), in

general, slightly increases with increasing the Bianchi parameter k. Changing the separationconstant, in general, keeps the overall structure of the radial function. However, increasingthe separation constant g leads to more oscillatory behaviour. In figure 3.5, we plot theradial functions, for k = 1

2and two other separation constants g = 6 and g = 12.

Superimposing all the different solutions with the different separation constants g, we canwrite the most general solutions to the partial differential equation (3.5) in D-dimensions,as

HD(r, x) = 1 +

∫ ∞0

dg

xN(P (g)IN(gx) +Q(g)KN(gx)

)R(r), (3.13)

where P (g) and Q(g) stand for the integration constants, for a specific value of the separationconstant g, and N is given by (3.9).

6

Figure 3.4: The numerical solutions for the radial function R(r), where k = 34

(up), k = 12

(middle) and k = 14

(down) with c = 1, g = 2.

Figure 3.5: The numerical solutions for the radial function R(r), where we set g = 6 (left)and g = 12 (right) with k = 1

2, c = 1.

To find the functions P (g) and Q(g), we may compare the general solutions (3.13), withanother related exact solutions to the theory. In fact, if we consider the large values for theradial coordinate r →∞, then the Bianchi type IX metric (2.8) changes to

dS2 = dr2 +r2

4(dθ2 + dφ2 + dψ2) +

r2 cos θ

2dφdψ, (3.14)

which is the metric on R4. Using the asymptotic metric (3.14) for the Bianchi type IXgeometry, we find an exact solutions to the Einstein-Maxwell theory in D-dimensions, where

7

the gravity is described by the metric

dS2D = − dt2

H(r, x)2+H(r, x)

2D−3 (dx2 + x2dΩ2

D−6 + dS2), (3.15)

together with the Maxwell’s field Fµν , as

Ftr = − α

H(r, x)2∂H(r, x)

∂r, (3.16)

Ftx = − α

H(r, x)2∂H(r, x)

∂x. (3.17)

We find that the metric function H in D-dimensions, is given by the exact form

HD(r, x) = 1 +γ

(r2 + x2)N+2, (3.18)

where γ is a constant.Finding the exact solutions (3.15) with (3.18), enable us to find an integral equation for

the functions P and Q in (3.13). In fact, we have the integral equation∫ ∞0

dg

xN(P (g)IN(gx) +Q(g)KN(gx)

)limr→∞

R(r) =γ

(r2 + x2)N+2. (3.19)

However, we know limr→∞R(r) is given by (3.12). Considering r1 = 1g, r2 = 0, we solve the

integral equation (3.19) and find

P (g) = 0, Q(g) =γ

2N+1Γ(N + 2)gN+3, (3.20)

where Γ(N + 2) is the gamma function. Moreover, considering the other possibility r1 = 0,does not lead to consistent solutions for the functions P and Q in the integral equation(3.19). Summarizing the results, we find the metric function in D-dimensions, as

HD(r, x) = 1 +γ

2N+1Γ(N + 2)

∫ ∞0

dg

xNgN+3KN(gx)R(r). (3.21)

4 The second class of solutions in D ≥ 6-dimensional

Einstein-Maxwell theory

In this section, we present the second independent class of solutions for the metric functionwhich satisfies the partial differential equation (3.5). In separation of the coordinates, wereplace g → ig in the differential equation (3.7), which its solutions, in terms of the Besselfunctions, read as

X(x) = x1JN(gx)

xN+ x2

YN(gx)

xN, (4.1)

8

where x1 and x2 are constants of integration, and N is given by (3.9). Moreover the radialdifferential equation becomes

(−256 c8k4 + 16 c4k4r4 + 16 c4r4 − r8

)r

d2

dr2R (r) +

(256 c8k4 + 16 c4k4r4 + 16 c4r4 − 3 r8

) d

drR (r)

+ g2r5√

256 c8k4 − 16 c4k4r4 − 16 c4r4 + r8R (r) = 0. (4.2)

We present the numerical solutions to equation (4.2) for 0 < k < 1, where its asymptoticform is given by

rd2

dr2R(r) + 3

d

drR(r)− rg2R(r) = 0. (4.3)

The exact solutions to equation (4.3) are given by

R(r) = r1I1(gr)

r+ r2

K1(gr)

r, (4.4)

where r1 and r2 are constants. In figure 4.1, we plot the behaviour of the R(r) where r →∞.We notice two very different behaviours for the asymptotic radial function, as r →∞.

Figure 4.1: The radial function R(r) for large values of r, where we set r1 = 1, r2 = 0 (left),and r1 = 0, r2 = 1 (right) and g = 1.

In figure 4.2, we plot the numerical solutions for the radial function R(r), where we setk = 1

2, c = 1 and g = 2. We notice from figure 4.2 that the radial function approaches zero

for r → 2c, and becomes large, as r →∞, in agreement with the asymptotic solutions (4.4),where r1 = 1, r2 = 0, and the left plot in figure 4.1.

9

Figure 4.2: The numerical solution for the radial function R(r), where we set k = 12, c = 1

and g = 2.

The general structure of the radial function is the same for the other values of the Bianchiparameter c. The divergent behaviour of the radial function happens at r → ∞ and theradial function approaches zero, for r → 2c.

Moreover, in figure 4.3, we plot the numerical solutions for the radial function R(r),where we set k = 1

4and k = 3

4with c = 1 and g = 2. Tough the figures 4.2 and 4.3 are quite

similar, however they have subtle dependence on the Bianchi parameter k. In figure 4.4, weplot three radial functions, over a small interval of r, for k = 1

4, 1

2and 3

4. As we notice from

figure 4.4, the radial function R(r), in general, slightly increases with increasing the Bianchiparameter k, very similar to the situation for the first radial solutions.

Changing the separation constant, in general, keeps the overall structure of the radialfunction. However, increasing the separation constant g leads to a slower decaying behaviourfor the radial function. In figure 4.5, we plot the radial functions, for k = 1

2and two other

separation constants g = 6 and g = 12.Hence, we find the second most general solutions for the metric function in D-dimensions,

as given by

HD(r, x) = 1 +

∫ ∞0

dg

xN(P (g)JN(gx) + Q(g)YN(gx)

)R(r), (4.5)

where P (g) and Q(g) stand for the integration constants, for a specific value of the separationconstant g and N is given by (3.9).

10

Figure 4.3: The numerical solutions for the radial function R(r), where we set k = 14

(left)and k = 3

4(right) with c = 1, g = 2.

Figure 4.4: The numerical solutions for the radial function R(r), where k = 34

(up), k = 12

(middle) and k = 14

(down) with c = 1, g = 2.

11

Figure 4.5: The numerical solutions for the radial function R(r), where we set g = 6 (left)and g = 12 (right) with k = 1

2, c = 1.

Requiring the metric function (4.5) in the limit of r →∞, reduces to the exact solutions(3.18), we find an integral equation for the functions P (g) and Q(g), as∫ ∞

0

dg

xN(P (g)JN(gx) + Q(g)YN(gx)

)limr→∞

R(r) =γ

(r2 + x2)N+2. (4.6)

Using equation (4.4) for limr→∞ R(r) with r1 = 1g, r2 = 0, we can solve the integral equation

(4.6) and find

P (g) =γΓ(−N)

2N+2(N + 1)gN+3, Q(g) = 0. (4.7)

Moreover, considering the other possibility r1 = 0, does not lead to consistent solutions forthe functions P and Q in the integral equation (4.6). Summarizing the results, we find thesecond metric function in D-dimensions, as

HD(r, x) = 1 +γΓ(−N)

2N+2(N + 1)

∫ ∞0

dg

xNgN+3JN(gx)R(r). (4.8)

5 The solutions in D ≥ 6 dimensional Einstein-Maxwell

theory with cosmological constant

In this section, we consider the D ≥ 6 Einstein-Maxwell theory with a cosmological constantΛ. We show that there are non-trivial solutions to the D ≥ 6 dimensional Einstein-Maxwelltheory with the cosmological constant. We start by considering the D ≥ 6 dimensioanlmetric, as

ds2D = −H(t, r, x)−2dt2 +H(t, r, x)1

N+2 (dx2 + x2dΩD−6 + ds2BIX), (5.1)

12

where the metric function depends on the coordinate t, as well as the coordinates r andx. The components of the Maxwell’s field, are given by equations (3.3) and (3.4), aftersubstitution H(r, x) to H(t, r, x). A lengthy calculation shows that all the Einstein’s andMaxwell’s field equations are satisfied by taking the separation of variables in the metricfunction, as

H(t, r, x) = T (t) +R(r)X(x), (5.2)

where the functions R(r) and X(x) satisfy exactly equations (3.10) and (3.7) for the firstclass of solutions, respectively. Of course the analytical continuation g → ig, yields thecorresponding equations for R(r) and X(x). Moreover, we find

T (t) = 1 + λt, (5.3)

where

λ = (D − 3)

√2Λ

(D − 2)(D − 1). (5.4)

Of course, we should consider only the positive cosmological constant. To summarize, wefind two classes of the cosmological solutions, where the metric functions are given by

HD(t, r, x) = 1 + λt+γ

2N+1Γ(N + 2)

∫ ∞0

dg

xNgN+3KN(gx)R(r), (5.5)

and

HD(t, r, x) = 1 + λt+γΓ(−N)

2N+2(N + 1)

∫ ∞0

dg

xNgN+3JN(gx)R(r). (5.6)

We note that the two metric functions (5.5) and (5.6) definitely approach to the exact metricfunction

HD(t, r, x) = 1 + λt+γ

(r2 + x2)N+2, (5.7)

in the limit of large radial coordinate. The metric function (5.7) is an exact solution to theEinstein’s and Maxwell’s field equations with a cosmological constant Λ, where

dS2D = − dt2

H(t, r, x)2+H(t, r, x)

2D−3 (dx2 + x2dΩ2

D−6 + dS2), (5.8)

where dS2 is given by (3.14).To get a glimpse of the solution (5.1), we consider the very late time coordinate, and find

ds2D = −dτ 2 + eλ

N+2τ (dx2 + x2dΩD−6 + ds2BIX), (5.9)

where

τ =ln(1 + λt)

λ' ln(λt)

λ. (5.10)

13

The constant-τ hypersurface of the metric (5.9) describes the foliation of (D−1)-dimensionsalde-Sitter space, by flat space-like Bianchi type IX. The volume of such a hyper-surface reachesits minimum at τ = 0, and increases exponentially with τ .

In figure 5.1, we plot the behaviour of the c-function for two different space-time dimen-sions. Both figures show increasing behaviour for the c-function which shows expansion ofthe constant-τ hyper-surface with the cosmological time τ . Moreover, we notice that in-creasing the dimension of spacetime leads to deceasing the value of the c-function. So, theconstant-τ hyper-surfaces expand slower in the higher dimensions.

Figure 5.1: The c-function of the space-time (5.9) versus the cosmological time coordinateτ for D = 6 (left) and D = 7 (right), where we set k = 1

2, c = 1, Λ = 4, r = 3.

6 The D ≥ 6 solutions with k = 0 and k = 1 in Einstein-

Maxwell theory

In previous sections, we constructed solutions toD ≥ 6-dimensional Einstein-Maxwell theory,with and without cosmological constant, based on Bianchi type IX geometry with 0 < k < 1.In this section, we consider such solutions, where k = 0 and k = 1.

6.1 k = 0

First we consider the Bianchi type IX metric (2.8) with k = 0. The metric (2.8) reduces to

ds2k=0 =dr2√1− a4

r4

+

√r4 − a4

4(σ2

1 + σ22) +

r2

4

σ23√

1− a4

r4

, (6.1)

where we set a = 2c and the radial coordinate r ≥ a. The metric (6.1) is the metric forthe Eguchi-Hanson type I geometry. The Ricci scalar for (6.1) is identically zero, and the

14

Kretschmann invariant is

K =384a8

(r2 + a2)3(r2 − a2)3, (6.2)

which indicates r = a is a singularity. The radial differential equation for R(r) is given by

r(a4 − r4) d2

dr2R (r) + (a4 − 3r4)

d

drR (r)− g2r3

√r4 − a4R (r) = 0, (6.3)

while the differential equation for X(x) is the same as (3.7) with the solutions (3.8). We findthe real analytical solutions for (6.3), which is

R(r) =1

2HC

(0, 0, 0,

ig2a2

2,−ig

2a2

4,a2 − i

√r4 − a4

2a2)

+1

2H∗C(0, 0, 0,

ig2a2

2,−ig

2a2

4,a2 − i

√r4 − a4

2a2), (6.4)

where HC is the Heun-C function. In figure 6.1, we plot the radial function for different valuesof the Eguchi-Hanson parameter a. We note that the radial function has the oscillatorybehaviour similar to cases where 0 < k < 1, however the radial function remains finite atr = a.

Figure 6.1: The radial function R(r) for k = 0, where we set a = 1 (left), and a = 2 (right)and g = 2.

Combining the different solutions for the radial function R(r) and X(x), we find the mostgeneral solution for the metric function HD(r, x) in D-dimensions, where k = 0, as

HD(r, x) = 1 +

∫ ∞0

dg

xN(P0(g)IN(gx) +Q0(g)KN(gx)

)× <

(HC

(0, 0, 0,

ig2a2

2,−ig

2a2

4,a2 − i

√r4 − a4

2a2)), (6.5)

15

where P0(g) and Q0(g) stand for the integration constants, for a specific value of the separa-tion constant g. To find the functions P0(g) and Q0(g), we consider the limit r →∞, wherethe Eguchi-Hanson type I space (6.1) becomes

ds2k=0 = dr2 +r2

4(σ2

1 + σ22 + σ2

3). (6.6)

The asymptotic line element (6.6) describes R4, and embedding it in the D-dimensionaltheory, leads to the exact solution (3.15)-(3.17) with the metric function (3.18). We find theintegral equation∫ ∞

0

dg

xN(P0(g)IN(gx) +Q0(g)KN(gx)

)limr→∞<(HC

(0, 0, 0,

ig2a2

2,−ig

2a2

4,a2 − i

√r4 − a4

2a2))

=γ

(r2 + x2)N+2, (6.7)

for the functions P0(g) and Q0(g). Moreover, we find

limr→∞

(HC

(0, 0, 0,

ig2a2

2,−ig

2a2

4,a2 − i

√r4 − a4

2a2))

=2

grJ1(gr). (6.8)

After a very long calculation, we find the solutions to the integral equation (6.7), as

P0(g) =γΓ(−N)

2N+3(N + 1)gN+3, Q0(g) = 0. (6.9)

Hence, we find the exact solutions for the metric function H(r, x) in D-dimensional Einstein-Maxwell theory with an embedded four-dimensional Eguchi-Hanson type I geometry (6.1)in the spatial section of the space-time, as

HD(r, x) = 1 +γΓ(−N)

2N+3(N + 1)

∫ ∞0

dg

xNgN+3IN(gx)<

(HC

(0, 0, 0,

ig2a2

2,−ig

2a2

4,a2 − i

√r4 − a4

2a2)).

(6.10)

We also verify that our numerical solutions to the differential equation (3.10), where k = 0,represent exactly the profile of the Heun-C function in equation (6.10).

Changing the separation constant g → ig generates the second class of solutions, wherek = 0. We find the radial equation

r(a4 − r4) d2

dr2R (r) + (a4 − 3r4)

d

drR (r) + g2r3

√r4 − a4R (r) = 0, (6.11)

where the exact solutions are given by

R(r) =1

2HC

(0, 0, 0,−ig

2a2

2,ig2a2

4,a2 − i

√r4 − a4

2a2)

+1

2H∗C(0, 0, 0,−ig

2a2

2,ig2a2

4,a2 − i

√r4 − a4

2a2). (6.12)

16

In figure 6.2, we plot the radial function R for different values of the Eguchi-Hanson param-eter a.

Figure 6.2: The radial function R(r) for k = 0, where we set a = 1 (left), and a = 2 (right)and g = 2.

We then superimpose the different solutions for the radial function R(r) and X(x), asgiven by (4.1), to find the second most general solution for the metric function HD(r, x) inD-dimensions, where k = 0, as

HD(r, x) = 1 +

∫ ∞0

dg

xN(P0(g)JN(gx) + Q0(g)YN(gx)

)× <

(HC

(0, 0, 0,−ig

2a2

2,ig2a2

4,a2 − i

√r4 − a4

2a2)), (6.13)

where P0(g) and Q0(g) stand for the integration constants, for a specific value of the separa-tion constant g. To find the functions P0(g) and Q0(g), we consider the limit r →∞, wherethe Eguchi-Hanson type I space (6.1) becomes (6.6). Hence we find the integral equation

∫ ∞0

dg

xN(P0(g)JN(gx) + Q0(g)YN(gx)

)limr→∞<(HC

(0, 0, 0,−ig

2a2

2,ig2a2

4,a2 − i

√r4 − a4

2a2))

=γ

(r2 + x2)N+2, (6.14)

for the functions P0(g) and Q0(g). We find

limr→∞

(HC

(0, 0, 0,−ig

2a2

2,ig2a2

4,a2 − i

√r4 − a4

2a2))

=2

grI1(gr). (6.15)

17

After a very long calculation, we find the solutions to the integral equation (6.14), as

P0(g) =(−1)N+1γΓ(−N − 1)

2N+3gN+3, Q0(g) = 0. (6.16)

Hence, we find the second exact solutions for the metric function H(r, x) in D-dimensionalEinstein-Maxwell theory with an embedded four-dimensional Eguchi-Hanson type I geometry(6.1) in the spatial section of the space-time, as

HD(r, x) = 1 +(−1)N+1γΓ(−N − 1)

2N+3

∫ ∞0

dg

xNgN+3JN(gx)<

(HC

(0, 0, 0,−ig

2a2

2,ig2a2

4,a2 − i

√r4 − a4

2a2)).

(6.17)

We also verify that our numerical solutions to the differential equation (4.2), where k = 0,represent exactly the profile of the Heun-C function in equation (6.17).

6.2 k = 1

In this section, we consider the Bianchi type IX metric (2.8) with k = 1. The metric (2.8)reduces to

ds2k=1 =dr2

1− a4

r4

+r4 − a4

4r2σ21 +

r2

4(σ2

2 + σ23), (6.18)

which is the known metric for the Eguchi-Hanson type II space, where r ≥ a. The Ricciscalar for (6.18) is identically zero, and the Kretschmann invariant is given by

K =384a8

r12, (6.19)

which is regular and finite, for r ≥ a.The exact solutions for the metric function H(r, x) in D-dimensional Einstein-Maxwell

theory with an embedded four-dimensional Eguchi-Hanson type II geometry (6.18) in thespatial section of the space-time, is given by [22]

HD(r, x) = 1 +γ

2ξD

∫ ∞0

dg

xNgN+3KN(gx)HC

(0, 0, 0,

−g2a2

2,g2a2

4,a2 − r2

2a2),

(6.20)

where ξ6+2n =√

π2(2n + 1)!!, n = 0, 1, 2, · · · for even dimensions D = 6 + 2n and ξ7+2n =

(2n + 2)!!, n = 0, 1, 2, · · · for odd dimensions D = 7 + 2n. Moreover, the second class ofsolutions, is given by the metric function

HD(r, x) = 1 +γπ(−)D

4ξD

∫ ∞0

dg

xNgN+3JN(gx)HC

(0, 0, 0,

g2a2

2,−g

2a2

4,a2 − r2

2a2).

(6.21)

We also verify that our numerical solutions to the differential equation (3.10), where k = 1,represent exactly the profile of the Heun-C functions in equations (6.20) and (6.21)

18

7 Concluding Remarks

We construct a class of exact solutions to the Einstein-Maxwell theory with a continuousparameter k. We find the metric function for the solutions in any dimensions D ≥ 6 uniquely,as a superposition of all radial solutions with their corresponding solutions in the other spatialdirection. We solve and present numerical solutions to the radial equation, as we can’t findany analytical solutions to the radial field equation. To find the weight functions in thesuperposition integral, we present another exact solutions to the Einstein-Maxwell theory,such that the superposition integral approaches to the exact metric function of the anothersolutions, in an appropriate limit. We find complicated integral equations for the weightfunctions, that we solve and find the unique solutions for the weight functions. We alsoconsider the positive cosmological constant, and show the field equations for the Einstein-Maxwell theory with positive cosmological constant, can be separated. We find the exactsolutions to the field equations and study the properties of the cosmological solutions. Wealso consider the special case, where the Bianchi type IX parameter k is zero. We showthat the Bianchi type IX geometry reduces to a less-known Eguchi-Hanson type I geometry.We find real analytical solutions to the radial equation, in any dimensions, in terms of theHeun-C functions. We also find the weight functions, such that the superimposed solutionsreduce to the known exact solutions, in an appropriate limit.

AcknowledgmentsThis work was supported by the Natural Sciences and Engineering Research Council of

Canada.

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